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On the plane problem in micropolar elasticity
Int. J. Ennng Sci., 1973, Vol.
11,pp.997-
1012.
Pergamon Press.
ON THE PLANE
Printed in Great Britain
PROBLEM IN MICROPOLAR ELASTICITY
B. M. CHIU and JAMES D. LEE School of Aeronautics, Astronautics, and Engineering Sciences, Purdue University, West Lafayette, Indiana47907, U.S.A.
Abstract-The most general solution of micropolar elasticity has been obtained for static plane problems in a circular ring-shaped region. For illustrative purpose, two examples of Volterra’s dislocation problem are given to demonstrate the physical meaning of many-valued solutions. 1. INTRODUCTION
SINCE micromorphic theory and its special case, micropolar theory, was formulated by Eringen and Suhubi[ l-31, only a few special problems have been solved. As far as micropolar elasticity is concerned, the following works are relevant to this work. Kaloni and Ariman [4] solved the stress concentration problem by modifying the results previously obtained by Mindlin [5,6]. Later Ariman [7] obtained additional results for plane problems of micropolar elasticity. Recently Parhi and Das[S] solved a problem of a circular plate being compressed diametrically for Mindlin’s couple-stress theory. In each of the above, solutions were obtained by utilizing the stress functions first introduced by Mindlin [6]. In this work we formulate the stress and displacement boundary value problems in cylindrical coordinates. Instead of working with stress functions, we deal directly with displacement fields and microrotation fields. We believe this direct approach enables us to develop a clear physical picture of the problem and eliminates the need for conjecture. In section 2 we formulate the static plane problem of micropolar elasticity with vanishing body forces and body couples into a set of three partial differential equations. The three unknowns are two displacements and one microrotation. The region of solution is a complete circular ring. In section 3 the most general single-valued solution for the displacement and microrotation fields is obtained. The stress functions corresponding to this general solution are obtained in section 4. We also demonstrate the physical meaning of many-valued solutions by giving two examples in section 5. These example problems are similar to Volterra’s dislocation problems in classical elasticity. We have utilized the results of section 3 to obtain these many-valued solutions. 2. BASIC EQUATIONS
To begin with, we recall the balance laws and constitutive equations which are basic to Eringen’s linear theory of micropolar elasticity[2] and relevant to the present work (static, no body forces and body couples):
mkl,k
•t
elm&n =
0
(2) (3)
mkl
=
adh,m8kl
+
P+k,l 997
UESVdllNo9-E
+
Y&k
(4)
998
B. M. CHIU
and JAMES
D. LEE
where akl and eklm are the Kronecker delta and the permutation symbol, respectively. In these equations, we employ a rectangular coordinate system and assume summation convention over repeated indices. The index following a comma represents partial differentiation with respect to a spatial coordinate. The variables &, &, f,& and mkl are the displacement, microrotation, stress tensor, and moment stress, respectively. The material moduli A, CL,k, a, p, and y are restricted, due to positive definiteness of internal energy, as follow: 3h+2p+k
a 0,
3a+/3+y
2p+k==O,
> 0,
-y
k>O
G p G y,
(5)
y > 0.
Substituting the constitutive equations (3) and (4) into the balance laws (1) and (2), we obtain the equations of motion in the following vector form: (h+2,u+k)VV.u-(~+k)VxVxu+kVx~=O
(6)
(a+P+y)VV.+yVxVx++k(Vxu-24)
=O.
We are interested in a plane problem with the displacement field and microrotation being specified in cylindrical coordinates (r, 0, z) as follow: U, = & = $0 = 0.
4, = +(r, 0)
U,‘U(T,0),
Ug z V(T, 6).
(7) field
(8)
Substituting equations (8) into equations (6), (7), we obtain a system of three linear partial differential equations with variable coefficients, (A+2p+k)(u,,,
+~LY,~-$U+~
1;,,,-$
V,e )
V,re+f
-_(p+k)(+.
(A+%++
V,e-$u,ee
+kSm.e=o )
(9)
V,ee+~U,re+~U.e)-k~,~
-(~+k)(--~,r~-~v,~+~v+~u,~e-~~,e)=O
(10)
~(9~rr+f~.~+~~,ee)+k(~,~+~~-~u,e-2d)=0.
The relevant physical components
of the stress tensor and the moment stress are
t,, = A u,,++++ (
tee=A
Ge =
(
u,,+fu+:v,e
ter= p
VW
v,e + (2p+k)u,,.
(12)
)
)
+(2p+k)
-+v+tu,e -~v+~u,~
CL VW
(11)
)
)
lu+lV (r r
.e)
+k(v,,-4)
+k
(
:U,e-:
(13) (14)
v++
)
(1%
On the plane problem in micropolar elasticity
999
mrz= Y4h
fl6)
1 ?&X2 = Y; cb,e.
(17)
We are looking for a solution in a circular ring-shaped region, i.e. (see Fig. 1) re[Riv
&]v
~~[O,
27T]
(18)
where the inner radius RImay be either zero or finite and the outer radius R. may be either finite or infinite. Since the ring is complete, we expect the displacements, microrotation, stress tensor, and moment stress to be single-valued. In other words, they are periodic functions of 8 with period 27r. The boundary conditions can be classified into two major categories: First boundary value problem
In this case, the surface tractions and surface couple are specified at the surfaces r = Ri and r = Ro, e.g. trr(Rr, 0) = N(e),
tre(Ri, 0) = S(fl), mr.z(Ri, 0) = n(d)*
(19)
Second boundary value problem
In this case, the displacements and microrotation are specified at the inner and outer surfaces, e.g. u(R$, 0) = D(@), v(Ri, 0) = ii(@),
#(R,, 19) = $(O).
m3
Fig. 1. Coordinate system and region of solution. 3. GENERAL
SOLUTION
Since the solution is periodic in 8 with period of 2rr, the set {eine;n = 0, Ifi:1, -C2, . . . .} forms a complete orthonormal basis. Thus we may expand u (r, 0) , u(r, 0), and #(r, 0) as follows: (cf. p. 96 of [9]): u(r,e)
=
5
U;I(r)eine ?Z=--m
(21)
1000
B. M. CHIU
and JAMES
D. LEE
m
$b(r.0) = E i@,(r)eine.
(23)
Substituting equations (21-23) into equations (9-l l), we obtain, for a particular n, a system of equations for the Fourier coefficients U,, V,, and @, (since there is no ambiguity, we may drop the subscript n) r2U,,,+rUIp--(l+A)U-rz(l-c)rV,,+n(l+c)V=nbr@ n(l-c)rU,,+n(l+c)U+c(fV,,,+rV,,)
(24) - (c+n”)V=
r’@,,,+&,,-nZ@-2d@
br%,,
= --(Tr(rV,r+ V-nU)
(25) (26)
where b = k/(h+2p+k), c = (p+k)/(A+2p+k),and(+ = k/y. Similarly, substituting equations (21-23) into equations (12-17), for a particular n, the Fourier coefficients of stress tensor and moment stress have the following expressions, t,.r=X
(
.,,++fV
)
+(2p+k)U.,
(27)
t,,=+.,++;V)+,,,+k,(+~V)
(28)
t,0 = i CL V,, +‘+;U)+k(V.,@)] i(
(29)
V+;U
)
+k ;U-+V+C’ (
mrz= iy@,,
(31)
m 0.z= -+.
(32)
To obtain the solution of U, V, and @, we observe that the left hand sides of equations (24), (25) are equidimensional, therefore we may introduce a new independent variable through the substitution r = e’. t = In r. (33) Consequently,
we have dZ d d” rz~+r~=dt2,r~=;li.
d
d (34)
Rewrite equations (24), (25) in the following form: o*-(l+rPc) [ n[(l-c)D+(l+c)]
-n[(l-c)DcD2-(c+n2)
(l+c)]][,]
= be’ [;;I
(35)
where D = (d/dt). It can be shown that equations equations (cf. p. 19 of [ lo]):
(35) are reduced to the following
[D-((n+l)][~-(-n+1)][D-(n-1)][D-(--n-l)]U=n~e~(D2--n2)~
(36)
1001
On the plane problem in micropolar elasticity
Now we look into three different cases: Case I: In/ 3 2
In this case, we first write the homogeneous solutions for U and I/, they are
To determine the relationships which must exist among the A’s and B’s, we introduce the expressions for U,, and V, into the left hand sides of equations (35) and require the results to be identically zero. Thus we find the following conditions B1 =
n+2-cn n-2c-cn
A,+
B
=; 2
n-2-cn -n-2c-kcn
A2 (3%
B4 = -A+
B, = As,
To find the particular solutions of equations (36), (37), we make use of the homogeneous solutions while employing the method of variation of parameters (cf. p. 25 of f 101).We write the particular solutions in the following form:
To find the a’s, we have the following condition 1 n-l -(nl- 1) (n- 1)2 (n- 1)2 -(n-1)3
(n-1)3 b =n;ef
r
After some manipulations we find al(t)
=
1 8(n+l)
be-n1(D+n)4, c
1 az(t) = S(n- 1) be”‘(D-n)Q, c as(t) = a*(t) = -
1 8(n-
be-(“-2)t[D+ (n-2)]@+g,(t)
1) c
1 b -etn+zH[D- (n+2)]@-gz(t) 8(n+ 1) c
(42)
1002
B. M. CHIU and JAMES D. LEE
where
I
t
e-(n-2W (s) ds (43)
ecn+z)s@(s) ds.
Similarly, we find for the b’s b,(r) = b,(t) = b3(f)=-
’
_be-nt[DP+ (n+2)0+2n]@ 1) c
’
_bent[DZ - (n-2)0-2n]@ 1) c
8n(n+ 8n(n-
be-‘n-2w~g2+nD-2n]~+g,(t)
’
8n(n-
b4(r) =-
1) c
be’“+*~t[D2-nD+2N]@+g~(t).
’
8n(n+
(44)
1) c
The general solutions for U and I/ are the sums U,, + UP and Vh+ VP, respectively, U(r) = AlP+‘+A*r-
n+l+A&-‘+Aqr-n--l rs-“+lQ,(S)
V(r) =
;T;c-cnA,r”“+_;I’,,s”;n T
+2(p+k)
v”-1
[
sP+QD
I
(s)
1
(45)
T sn+l@(s) ds
1.
(46)
I
Azr-“+I +A/-l
k
ds
‘.P+l@(s)
ds-,.-n-1
--A,~-“-’
ds + Pm1
I
Now we substitute equations (49, (46) into equation (26) and get the governing equation for @, rQ-D,,,+FD,,-
(n2+6*r2)@= -V
4(n+ l)A#+*+ [ n-2c-cn
where 6* = k (2~ + k) /y ( p + k). It is straightforward Cp( r) , which is Q(r) = ASK,(h)
+A,Z,(ar)
+
p+k 2p+k
1
4(n- l)A2 r-n+2 -n-2c+ cn
to get the general solution for
4(n+ l)A, P+
4(n-l)A2
Tn
-n-2c+cn
[ n-2c-cn
where K, (Sr) and Z,(&-) are the modified Bessel functions equation (48) into equations (49, (46), we obtain U(r) =
(47)
1
(48)
of order It. Substituting
1+ &n-~-cn]A”“il+[l+(2p+k)(~~2c-cn)]A2r-’i1 +AaP-1+AIrp”-1+2a(p+k)
k
{AS [K,+, (ar) -KM
-4
(6r)
1
[Z,+ltar) - Ll tar) II
(49)
1003
On the plane problem in micropolar elasticity
V(r)
= (n+2-cn)(2p+k)+ (n+2)kAlp+‘+ (2p+k) (n-2c-cn)
x Ap+1
+A#-’
-&-n--l
+
(n-2-m)(2p+k)+ (2p+k)(-n-2c+cn)
k (
2*
~
+
(n-2)k
{A:, [--Kn+1(G-1- zL(6r)
k)
I
+4irL+l(~r) + In-1car)II-
(50)
n=kl In this case we have
Case2:
The homogeneous
0’(0+2)(0-2)U=+e’(D2-l)Q,
(51)
D2(D+2)(D-2)V=8e1(D3+2D2-D-2)@.
(52)
solutions are U,(t)=A,
vh(f)
(53)
+A3t+A3e2f+A4e-2t
=,(1++4,+(1-d42+A
t+
1+c
-
2
3-c 1_3cA3
ezt TA, ee2’.
(54)
The particular solutions are written in the following form: (55) Again employing the method of variation of parameters,
we find
al(t) =~~[tec~~-(ec+tec)~]~j,(t)
a,(t) =
T$ [e’(D-
l)@]
a,(t) = +
[e+(D+
u,(t) = 3&
[e3c(D-3)@]
b,(t) =&
l)@]
[tecD2~+((tet-ec)D~-22tec~]+j,(t)
b2(t> =-$
[et(D2+D-2)@]
b3(f) =&
[e-t(D2+3D+2)Q]
b(f)
=-&
Tj,(r)
[e3t(D2-D+2)
(56)
1004
B. M. CHIU
and JAMES
D. LEE
where
(57)
Consequently,
the general solutions for U and I/ can be written as
U(r) =A,+Azlnr+A,r2+Aq~-2~
k 2(p+k)
(l-c)A,+A V(r) = & (l+cM,+ 1+c [
2
Inr+
+ (I 3-c m
A/
k
r
+2(p+k)
Q(s) ds-r+
(I
- A,re2
Q(s) ds+re2
i’ s2Q,(s) ds)
(58)
[
(59)
I
s’@(s) ds).
Substituting equations (58), (59) into equation (26), we obtain r2@,TT+r@,I.-
(1 +62r2)@ = Tcr &A2r+ [
&A/].
(60)
Finally we get Q(r) = A,K, (Sr) +A6Z1(8r) * 6 U(r)
=AI-
[&A,+
k
k [ ‘+(2p+k)(l+c)
2(2p+k)(1+c)A2+
1 X{A,[K,(W -&(Wl --AJZ2(W -b(WlI
+ [ ‘+
(2p+k:(l-3c)
+&A,r]
(61)
1
A,lnr
A,rZ+A4r-2i2(p:k)8
(62)
‘cr) =~~,+~[1-“+2(2~+k~]+[1+(2~+k:(1+c~]A21nr +A, l-3c
[
3-c+
X FAJK,(W Case3:
&]
P-A4r-2j+
2S(;+k)
+Ko(Wl+&[~2(W +~o(Wl~
(63)
n=O
In this case we have U(r) = A,r+A,r-’ V(r) =A,r+A4r-1+g(p+k) (a(r) = A,+A,Ko(6r)
(64) k
[-A,KI(Gr)+A6Z1(6r)]
+A,Z,(&).
(65) (66)
1005
On the plane problem in micropolar elasticity
The Fourier coefficients of stress tensor and moment stress are found to be
t,,={(n+1)(2p+k)[-4ch+2(p+k)n--(n+2)(2~+k)])&~ +((n-1)(2~+k)[-4ch-22(~+~k)n+c(n-22)(2CL+k)l}B,r-~ + (2/&+-t) (n-&.!~~p>~, --As[(n+ to,=
l)A,r”-2-
(2p.k)
{k&J (n+ l)&+,(W 1)1,+,(6r) + b-
(n+ 1)&-n--2 + (n-
1)~,-1(6r)l)
1) (2p+k)[-4~h-2n(p++)
{(n+
~)K-I(WI
+c(n-2)
(2p+k)])&P
+{(~-1)(2~+k)[-4~h+2n(p+k)-c(n+2)(2p+k)]~&~-~ - (2p.k) +&!$:T)
-A,[
(n-
1)A,P-2+ {&[(n+
(n+ l)Aqren-2
(2p.k)
+ (n-l)L1(Wl
l)K,+I(6r)
(n+ 1)~,+,(W + (n- l)I,-,(Wl)
tre = i n(2p+k)[2(p+k)
-c(2p+k)][ (n+ l)BIP+ 1 + (n-l>(2~.~+k)A~~-~+(n+1)(2~+$_)A,r-”-~ +&($p:y) -M(n+
[&I(~+
{
[ML(W
- (n-l)K&WI
+hJ,(W
1
4n(~+k)[(n+l)B,r”-‘+(n-l)B,r-“-‘I -i
m
l)B,r-“1
(~-l)~n-I(Wll}
l)Zn+@)--
t&.= t,,+ik$y mrz=iy
l)&+I(i+)
(n-
k(ZG+l(~r)
+ZL(&))
-A,(Z,+,(&)
+I.l(sr))]}
4n(~+k)[(n+l)BI~-1-(n-l)B,r-n-1]+~[A,K,(Gr)+A,Z,(6r)]
(67)
where
t?T=
2cA+2(p+k)
+c(2p+k) 1+c
-2(2~+k)A,r-3S~~~~=~~
A r_,+-8cA+4(p+k) 2
{A,&(&)
-6c(2p+k) I-3c
--A,Z,(Sr)}
A r 3
1006
B. M. CHIU
t*e=2ch-2(/.h+k)+c(2p+k)A l+C i2(2p-t-k)A4r-3* tro = A i
c(2 I
and JAMES
D. LEE
r_,+-8cX-4(p+k)-2c(2p+k)A 2 l-3c ~~~~~~~ wG(Sr)
-tk) 4(p+k) l’+ c &r-l -t
r 3
-N,(Sr)I
-2c(2p+k) l-3c
/g r+2(2p+k)A 3
4
r-3 I
m et=-Y
(68)
t2=0 trr=
(2hf2,~+k)A,-(2p+k)A~r~~
tee= (2h-b2p+k)AI+
(2p+k)A,re2
tM = i - (2p.+k)A4r++ II (Sr) -r-f-
ter = i - (2~ + k)A4rp2 +
172, =
~y{S[--~5~dSr)
lo(Sr)
+4J@r)l}
m @r--0
(69) 4. STRESS
FUNCTIONS
Mindlin[5,6] introduced two stress functions for his couple-stress theory in the same fashion as the Airy stress function was introduced in classical elasticity. His fo~ulation can be modified slightly to fit the micropolar theory (cf. Kaloni and Ariman [4]). We now reproduce Mind&r’s results in our notations. The relations between stress functions and components of stress and moment stress are (70) (71) (72)
On the plane problem in micropolar elasticity
1007
ter =I -iF,t-e+$F,e+Gw
(731
mrr= GW
(74)
I G,e.
mb=;
(75)
And the governing equations for the stress functions are (761 (77)
V4F=#,V2
G-~,G)=O. C
(78)
The general solutions for F and G, co~esponding to the soluti~us found in the previous section, are
Case2:
n=+l
F = ekze (2pfk)A,r’+
CaseJ:
&A*[
(*p-i-k)
(-rlnr)
-2i(h+2p+k)rOl
n=O G = iy[A,K,(6r)
F=$
(2h+2p+k)AIrZ
5. EXCEPTIONAL
+A,I,(6r)]
(83)
-(2pfk)A21nr-i(2p+k)A&
034)
CASES-VOLTERRA’S
DISLOCATIONS
In the previous sections, we have formulated the static problem of micropolar elasticity in cylindrical coordinates and obtained general solutions for boundary value
1008
B. M. CHIU
and JAMES
D. LEE
problems of the first and second kinds in a circular ring-shaped region. However those solutions are only valid for single-valued displacement and microrotation fields. In this section, we demonstrate the physical meaning of many-valued solutions by giving the following two examples: Example
1
We cut a ring (see Fig. 2) and apply forces P until an initial displacement d is obtained. Then we rejoin the two ends by welding or other means. A ring with initial stresses is thus obtained. P
I
Fig. 2. Dislocation of ring with parallel fissure.
We try the following set of solutions for this problem, r/G-,@ =G v(r,fQ =G c$(r,O)
d
sin 0 -ecoso+4(h+2P+k)
d
cos 8 ~sin~+4(h+2P+k)
[2(2~++) [2(2p+k)
lnrlnr+
(2X+2~+$_)] (2h+21.~+k)]
(85) (86)
=$yc
Substituting equations (85-87) into equations (9-l l), we realize that the equations of motion are satisfied identically. From equations (12- 17), we obtain t,,=
Iss=*sine 2~
h=f0r=-~
mrz =m0.:=-y--
G%+$-~)(~A+~P+N 2(h+2/_+k)
r
d cosO (2/.~+k) (2h+2p+k) r
2(h+2p+k)
d cos 0 2rr r2 .
In order to have stress free boundaries, i.e. t,, = trO= m, = 0 at r = R,, and r = Ri, another set of solutions (making use of the results obtained in section 3)
we superpose
1009
On the plane problem in inicropolar elasticity
and we get u(r,8)
=gsinO
2h-2 -k 2p fk
B8 + BP
1 1 2(2p+kk) +2p+k
II Br2_B,__Z 1 tr(r,O) =&se i6A+5(2P+k) x B Kt(Sr) -&(‘r) I 5 2
_B
6
Zz(6r) -Zo(sr) 2
a
4
--
2p.+k
x B 5 &(‘r) i
+&far) 2
2(2p+k)
_B 6 zdSr) +f,t6r) 2
(89)
1 2p+k
I}
(90)
where B3 = M/N B = (2p+k) (2h+2p+k) 4 2(h+2,u+k) -
Rf_2(2h+2p+k)
& (S&l Wm -R&J I+
~6~(~+2~+k)~ (2p+k)SN
x [ I2 (SR,) Rf - Zz(SR,) R:] } + N = 2(2h-t2p+k)S(R~--RI)
~Rf+~[Zp(6Ri)(R19*-R:h) R% & (SRd (rlz - 52)
(2~+k)~2~+2~+k) 2(h+2p+k)
- ‘6y(:;f:~+k)
ScRz_RR2)
’ { ([1-ql)
O
[Zz(&Ri)Rf
51 G Kz(SRi) +Ko(6Ri) 362 5 Zz(SRi) +Zo(6Ri) rll = &(SRo)
+ Ko(6Ro) t 72 = Zz(6R,,) + Z,(6Ro).
In this case a portion of the ring between two adjacent cross sections is cut out (see Fig. 3). The ends of the ring are joined again by welding or other means. We again obtain a ring with initial stresses.
1010
B. M. CHIU
and JAMES
D. LEE
Fig. 3. Dislocation of ring with radial fissure.
For this problem we try the following set of solutions: 2p+k
“Cry e,=4n(X+2p+k)
vlnr
(92)
U(T, e) =+I $(r, e)
(93)
=&e.
Again we find that this set of solutions satisfies equations (9- 11). Correspondingly, obtain
2h+2p-+k (2p+k)2(h+2p+k)
h(2E.c+k)
lnr+2(A+2p+k)
we
+A+2p+k
fro = ter = mrz = 0
m BZ
=zf;;y+.
(95)
Making use of the results obtained in section 3, we superpose another set of solutions: U(T, 0) = AIr+Az~-l,
u(r, 0) = $(r, e) = 0.
(96)
By taking
(97)
On the plane problem in micropolar elasticity
1011
we have made t,, = fro= 171,= 0 at r = R,, and r = Ri. It is interesting to note that this ring has the following initial stresses: E (2/L+k)(2h+2p+k) tee = G h+2/_L+k
(2+ln(&)+~ln(~)[R~(l-$!
+R;(l-$)I} (98) These stresses are B-independent. 6. CONCLUSION
The main purpose of the present work is to solve the boundary value problem of micropolar elasticity in a complete ring-shaped region. At the boundaries r = Ri and r = Ro, either displacements and microrotation U, o, + or stress and moment stress components trr, rM, m,, are specified as functions of 8. For single-valued solutions, U, u, and 4 are periodic in 8 with period of 2~. Therefore the set {efne;n = 0, n = + 1, n = 52,. . . .} forms a complete orthonormal basis. We then reduce the problem to a system of three simultaneous ordinary differential equations of second order with the Fourier coefficients U,(r), V,(r), and an(r) being the unknowns. Finally we obtain the most general solutions of U,, V,, and @‘nfor every n with six arbitrary constants. The remaining task for each boundary value problem is to seek the Fourier coefficients of given boundary conditions and then determine the corresponding six constants according to the given boundary conditions. We also give two examples to show the physical meaning of many-valued solutions and illustrate how the corresponding single-valued solutions are being utilized. Acknowledgement-The second author (JDL) is indebted to the support by a grant, No. GK-20657, from National Science Foundation. REFERENCES [l] A. C. ERINGEN and E. S. SUHUBI, Inr. J. Engng Sci. 2,189 (1964). [2] A. C. ERINGEN,J. Math. Mech. 15,909 (1966). [3] A. C. ERINGEN,J. Math. Mech. 16,1(1966). [4] P. K. KALONI and T. ARIMAN,ZAMP 17,136 (1967). [5] R. D. MINDLIN and H. F. TIERSTEN,Arch. ration Mech. Anafysisll, 415, (1962). [6] R. D. MINDLIN, Exp. Mech. 3,1(1963). [7] T. ARIMAN,Acta Mech. 4,216 (1967). [8] H. K. PARHI and A. K. DAS, Bull. Acnd. Pofon. Sci.,Ser. Sci. Tech. 19,13 1 (1971). 191 R. V. CHURCHILL, Fourier Series and Boundary Value Problems. McGraw-Hill (1963). 1101F. B.HILDEBRAND,Aduanced CafculusforAppfications. Prentice-Hall (1962). (Received 14 September 1972) Resume- La solution la plus genkale de I’ClasticitC micropolarie a &tCobtenue pour des problemes statiques plans darts une region circulaire en forme d’anneau. A titre d’illustration, deux exemples de probltme de dislocation de Volterra sont don&s pour montrer la signification physique des solutions a plusieurs valeurs. Zusammenfassung - Die allgemeinste L&sung mikropolarer Elastixitit wurde fiir statische ebene Probleme in einer runden ringfkmigen Zone erhalten. Zur Verbildlichung werden xwei Beispiele von Volterra’s Verdr@tmgsproblem gegeben, urn die physikalische Bedeutung mehrfach-bewerteter Losungen nachzuweisen.
1012
B. M. CHIU
and JAMES
D. LEE
Sommario - La soluzione pili generale dell’elasticit8 micropolare t stata ricavata nei riguardi di problemi di piano static0 in una regione circolare a forma di cerchio. A scopi illustrativi, si danno due esempi de1 problema di spostamento di Volterra per dimostrare il significato fisico delle soluzioni a pi6 valori. A~IZT~~KT - nonyreHo Hau6onee 06mee pemeHue ME~K~OIIOJIX~HO~~ YIIPY~OCTH AJIHCTaTn’IecKnx ~OCKAX npo6neM B KOJIbueO6pa3HOiio6nacTn. B qeJuix HarJt~AHOCTA,naIOTCnnBe npuMepa npO6neMbI ~ncnoKaqm% BOJlbTeppbl nOKa3aTb @i3WieCKHi CMblCJIMHOr03HaYHbIX npo6neiw
11,pp.997-
1012.
Pergamon Press.
ON THE PLANE
Printed in Great Britain
PROBLEM IN MICROPOLAR ELASTICITY
B. M. CHIU and JAMES D. LEE School of Aeronautics, Astronautics, and Engineering Sciences, Purdue University, West Lafayette, Indiana47907, U.S.A.
Abstract-The most general solution of micropolar elasticity has been obtained for static plane problems in a circular ring-shaped region. For illustrative purpose, two examples of Volterra’s dislocation problem are given to demonstrate the physical meaning of many-valued solutions. 1. INTRODUCTION
SINCE micromorphic theory and its special case, micropolar theory, was formulated by Eringen and Suhubi[ l-31, only a few special problems have been solved. As far as micropolar elasticity is concerned, the following works are relevant to this work. Kaloni and Ariman [4] solved the stress concentration problem by modifying the results previously obtained by Mindlin [5,6]. Later Ariman [7] obtained additional results for plane problems of micropolar elasticity. Recently Parhi and Das[S] solved a problem of a circular plate being compressed diametrically for Mindlin’s couple-stress theory. In each of the above, solutions were obtained by utilizing the stress functions first introduced by Mindlin [6]. In this work we formulate the stress and displacement boundary value problems in cylindrical coordinates. Instead of working with stress functions, we deal directly with displacement fields and microrotation fields. We believe this direct approach enables us to develop a clear physical picture of the problem and eliminates the need for conjecture. In section 2 we formulate the static plane problem of micropolar elasticity with vanishing body forces and body couples into a set of three partial differential equations. The three unknowns are two displacements and one microrotation. The region of solution is a complete circular ring. In section 3 the most general single-valued solution for the displacement and microrotation fields is obtained. The stress functions corresponding to this general solution are obtained in section 4. We also demonstrate the physical meaning of many-valued solutions by giving two examples in section 5. These example problems are similar to Volterra’s dislocation problems in classical elasticity. We have utilized the results of section 3 to obtain these many-valued solutions. 2. BASIC EQUATIONS
To begin with, we recall the balance laws and constitutive equations which are basic to Eringen’s linear theory of micropolar elasticity[2] and relevant to the present work (static, no body forces and body couples):
mkl,k
•t
elm&n =
0
(2) (3)
mkl
=
adh,m8kl
+
P+k,l 997
UESVdllNo9-E
+
Y&k
(4)
998
B. M. CHIU
and JAMES
D. LEE
where akl and eklm are the Kronecker delta and the permutation symbol, respectively. In these equations, we employ a rectangular coordinate system and assume summation convention over repeated indices. The index following a comma represents partial differentiation with respect to a spatial coordinate. The variables &, &, f,& and mkl are the displacement, microrotation, stress tensor, and moment stress, respectively. The material moduli A, CL,k, a, p, and y are restricted, due to positive definiteness of internal energy, as follow: 3h+2p+k
a 0,
3a+/3+y
2p+k==O,
> 0,
-y
k>O
G p G y,
(5)
y > 0.
Substituting the constitutive equations (3) and (4) into the balance laws (1) and (2), we obtain the equations of motion in the following vector form: (h+2,u+k)VV.u-(~+k)VxVxu+kVx~=O
(6)
(a+P+y)VV.+yVxVx++k(Vxu-24)
=O.
We are interested in a plane problem with the displacement field and microrotation being specified in cylindrical coordinates (r, 0, z) as follow: U, = & = $0 = 0.
4, = +(r, 0)
U,‘U(T,0),
Ug z V(T, 6).
(7) field
(8)
Substituting equations (8) into equations (6), (7), we obtain a system of three linear partial differential equations with variable coefficients, (A+2p+k)(u,,,
+~LY,~-$U+~
1;,,,-$
V,e )
V,re+f
-_(p+k)(+.
(A+%++
V,e-$u,ee
+kSm.e=o )
(9)
V,ee+~U,re+~U.e)-k~,~
-(~+k)(--~,r~-~v,~+~v+~u,~e-~~,e)=O
(10)
~(9~rr+f~.~+~~,ee)+k(~,~+~~-~u,e-2d)=0.
The relevant physical components
of the stress tensor and the moment stress are
t,, = A u,,++++ (
tee=A
Ge =
(
u,,+fu+:v,e
ter= p
VW
v,e + (2p+k)u,,.
(12)
)
)
+(2p+k)
-+v+tu,e -~v+~u,~
CL VW
(11)
)
)
lu+lV (r r
.e)
+k(v,,-4)
+k
(
:U,e-:
(13) (14)
v++
)
(1%
On the plane problem in micropolar elasticity
999
mrz= Y4h
fl6)
1 ?&X2 = Y; cb,e.
(17)
We are looking for a solution in a circular ring-shaped region, i.e. (see Fig. 1) re[Riv
&]v
~~[O,
27T]
(18)
where the inner radius RImay be either zero or finite and the outer radius R. may be either finite or infinite. Since the ring is complete, we expect the displacements, microrotation, stress tensor, and moment stress to be single-valued. In other words, they are periodic functions of 8 with period 27r. The boundary conditions can be classified into two major categories: First boundary value problem
In this case, the surface tractions and surface couple are specified at the surfaces r = Ri and r = Ro, e.g. trr(Rr, 0) = N(e),
tre(Ri, 0) = S(fl), mr.z(Ri, 0) = n(d)*
(19)
Second boundary value problem
In this case, the displacements and microrotation are specified at the inner and outer surfaces, e.g. u(R$, 0) = D(@), v(Ri, 0) = ii(@),
#(R,, 19) = $(O).
m3
Fig. 1. Coordinate system and region of solution. 3. GENERAL
SOLUTION
Since the solution is periodic in 8 with period of 2rr, the set {eine;n = 0, Ifi:1, -C2, . . . .} forms a complete orthonormal basis. Thus we may expand u (r, 0) , u(r, 0), and #(r, 0) as follows: (cf. p. 96 of [9]): u(r,e)
=
5
U;I(r)eine ?Z=--m
(21)
1000
B. M. CHIU
and JAMES
D. LEE
m
$b(r.0) = E i@,(r)eine.
(23)
Substituting equations (21-23) into equations (9-l l), we obtain, for a particular n, a system of equations for the Fourier coefficients U,, V,, and @, (since there is no ambiguity, we may drop the subscript n) r2U,,,+rUIp--(l+A)U-rz(l-c)rV,,+n(l+c)V=nbr@ n(l-c)rU,,+n(l+c)U+c(fV,,,+rV,,)
(24) - (c+n”)V=
r’@,,,+&,,-nZ@-2d@
br%,,
= --(Tr(rV,r+ V-nU)
(25) (26)
where b = k/(h+2p+k), c = (p+k)/(A+2p+k),and(+ = k/y. Similarly, substituting equations (21-23) into equations (12-17), for a particular n, the Fourier coefficients of stress tensor and moment stress have the following expressions, t,.r=X
(
.,,++fV
)
+(2p+k)U.,
(27)
t,,=+.,++;V)+,,,+k,(+~V)
(28)
t,0 = i CL V,, +‘+;U)+k(V.,@)] i(
(29)
V+;U
)
+k ;U-+V+C’ (
mrz= iy@,,
(31)
m 0.z= -+.
(32)
To obtain the solution of U, V, and @, we observe that the left hand sides of equations (24), (25) are equidimensional, therefore we may introduce a new independent variable through the substitution r = e’. t = In r. (33) Consequently,
we have dZ d d” rz~+r~=dt2,r~=;li.
d
d (34)
Rewrite equations (24), (25) in the following form: o*-(l+rPc) [ n[(l-c)D+(l+c)]
-n[(l-c)DcD2-(c+n2)
(l+c)]][,]
= be’ [;;I
(35)
where D = (d/dt). It can be shown that equations equations (cf. p. 19 of [ lo]):
(35) are reduced to the following
[D-((n+l)][~-(-n+1)][D-(n-1)][D-(--n-l)]U=n~e~(D2--n2)~
(36)
1001
On the plane problem in micropolar elasticity
Now we look into three different cases: Case I: In/ 3 2
In this case, we first write the homogeneous solutions for U and I/, they are
To determine the relationships which must exist among the A’s and B’s, we introduce the expressions for U,, and V, into the left hand sides of equations (35) and require the results to be identically zero. Thus we find the following conditions B1 =
n+2-cn n-2c-cn
A,+
B
=; 2
n-2-cn -n-2c-kcn
A2 (3%
B4 = -A+
B, = As,
To find the particular solutions of equations (36), (37), we make use of the homogeneous solutions while employing the method of variation of parameters (cf. p. 25 of f 101).We write the particular solutions in the following form:
To find the a’s, we have the following condition 1 n-l -(nl- 1) (n- 1)2 (n- 1)2 -(n-1)3
(n-1)3 b =n;ef
r
After some manipulations we find al(t)
=
1 8(n+l)
be-n1(D+n)4, c
1 az(t) = S(n- 1) be”‘(D-n)Q, c as(t) = a*(t) = -
1 8(n-
be-(“-2)t[D+ (n-2)]@+g,(t)
1) c
1 b -etn+zH[D- (n+2)]@-gz(t) 8(n+ 1) c
(42)
1002
B. M. CHIU and JAMES D. LEE
where
I
t
e-(n-2W (s) ds (43)
ecn+z)s@(s) ds.
Similarly, we find for the b’s b,(r) = b,(t) = b3(f)=-
’
_be-nt[DP+ (n+2)0+2n]@ 1) c
’
_bent[DZ - (n-2)0-2n]@ 1) c
8n(n+ 8n(n-
be-‘n-2w~g2+nD-2n]~+g,(t)
’
8n(n-
b4(r) =-
1) c
be’“+*~t[D2-nD+2N]@+g~(t).
’
8n(n+
(44)
1) c
The general solutions for U and I/ are the sums U,, + UP and Vh+ VP, respectively, U(r) = AlP+‘+A*r-
n+l+A&-‘+Aqr-n--l rs-“+lQ,(S)
V(r) =
;T;c-cnA,r”“+_;I’,,s”;n T
+2(p+k)
v”-1
[
sP+QD
I
(s)
1
(45)
T sn+l@(s) ds
1.
(46)
I
Azr-“+I +A/-l
k
ds
‘.P+l@(s)
ds-,.-n-1
--A,~-“-’
ds + Pm1
I
Now we substitute equations (49, (46) into equation (26) and get the governing equation for @, rQ-D,,,+FD,,-
(n2+6*r2)@= -V
4(n+ l)A#+*+ [ n-2c-cn
where 6* = k (2~ + k) /y ( p + k). It is straightforward Cp( r) , which is Q(r) = ASK,(h)
+A,Z,(ar)
+
p+k 2p+k
1
4(n- l)A2 r-n+2 -n-2c+ cn
to get the general solution for
4(n+ l)A, P+
4(n-l)A2
Tn
-n-2c+cn
[ n-2c-cn
where K, (Sr) and Z,(&-) are the modified Bessel functions equation (48) into equations (49, (46), we obtain U(r) =
(47)
1
(48)
of order It. Substituting
1+ &n-~-cn]A”“il+[l+(2p+k)(~~2c-cn)]A2r-’i1 +AaP-1+AIrp”-1+2a(p+k)
k
{AS [K,+, (ar) -KM
-4
(6r)
1
[Z,+ltar) - Ll tar) II
(49)
1003
On the plane problem in micropolar elasticity
V(r)
= (n+2-cn)(2p+k)+ (n+2)kAlp+‘+ (2p+k) (n-2c-cn)
x Ap+1
+A#-’
-&-n--l
+
(n-2-m)(2p+k)+ (2p+k)(-n-2c+cn)
k (
2*
~
+
(n-2)k
{A:, [--Kn+1(G-1- zL(6r)
k)
I
+4irL+l(~r) + In-1car)II-
(50)
n=kl In this case we have
Case2:
The homogeneous
0’(0+2)(0-2)U=+e’(D2-l)Q,
(51)
D2(D+2)(D-2)V=8e1(D3+2D2-D-2)@.
(52)
solutions are U,(t)=A,
vh(f)
(53)
+A3t+A3e2f+A4e-2t
=,(1++4,+(1-d42+A
t+
1+c
-
2
3-c 1_3cA3
ezt TA, ee2’.
(54)
The particular solutions are written in the following form: (55) Again employing the method of variation of parameters,
we find
al(t) =~~[tec~~-(ec+tec)~]~j,(t)
a,(t) =
T$ [e’(D-
l)@]
a,(t) = +
[e+(D+
u,(t) = 3&
[e3c(D-3)@]
b,(t) =&
l)@]
[tecD2~+((tet-ec)D~-22tec~]+j,(t)
b2(t> =-$
[et(D2+D-2)@]
b3(f) =&
[e-t(D2+3D+2)Q]
b(f)
=-&
Tj,(r)
[e3t(D2-D+2)
(56)
1004
B. M. CHIU
and JAMES
D. LEE
where
(57)
Consequently,
the general solutions for U and I/ can be written as
U(r) =A,+Azlnr+A,r2+Aq~-2~
k 2(p+k)
(l-c)A,+A V(r) = & (l+cM,+ 1+c [
2
Inr+
+ (I 3-c m
A/
k
r
+2(p+k)
Q(s) ds-r+
(I
- A,re2
Q(s) ds+re2
i’ s2Q,(s) ds)
(58)
[
(59)
I
s’@(s) ds).
Substituting equations (58), (59) into equation (26), we obtain r2@,TT+r@,I.-
(1 +62r2)@ = Tcr &A2r+ [
&A/].
(60)
Finally we get Q(r) = A,K, (Sr) +A6Z1(8r) * 6 U(r)
=AI-
[&A,+
k
k [ ‘+(2p+k)(l+c)
2(2p+k)(1+c)A2+
1 X{A,[K,(W -&(Wl --AJZ2(W -b(WlI
+ [ ‘+
(2p+k:(l-3c)
+&A,r]
(61)
1
A,lnr
A,rZ+A4r-2i2(p:k)8
(62)
‘cr) =~~,+~[1-“+2(2~+k~]+[1+(2~+k:(1+c~]A21nr +A, l-3c
[
3-c+
X FAJK,(W Case3:
&]
P-A4r-2j+
2S(;+k)
+Ko(Wl+&[~2(W +~o(Wl~
(63)
n=O
In this case we have U(r) = A,r+A,r-’ V(r) =A,r+A4r-1+g(p+k) (a(r) = A,+A,Ko(6r)
(64) k
[-A,KI(Gr)+A6Z1(6r)]
+A,Z,(&).
(65) (66)
1005
On the plane problem in micropolar elasticity
The Fourier coefficients of stress tensor and moment stress are found to be
t,,={(n+1)(2p+k)[-4ch+2(p+k)n--(n+2)(2~+k)])&~ +((n-1)(2~+k)[-4ch-22(~+~k)n+c(n-22)(2CL+k)l}B,r-~ + (2/&+-t) (n-&.!~~p>~, --As[(n+ to,=
l)A,r”-2-
(2p.k)
{k&J (n+ l)&+,(W 1)1,+,(6r) + b-
(n+ 1)&-n--2 + (n-
1)~,-1(6r)l)
1) (2p+k)[-4~h-2n(p++)
{(n+
~)K-I(WI
+c(n-2)
(2p+k)])&P
+{(~-1)(2~+k)[-4~h+2n(p+k)-c(n+2)(2p+k)]~&~-~ - (2p.k) +&!$:T)
-A,[
(n-
1)A,P-2+ {&[(n+
(n+ l)Aqren-2
(2p.k)
+ (n-l)L1(Wl
l)K,+I(6r)
(n+ 1)~,+,(W + (n- l)I,-,(Wl)
tre = i n(2p+k)[2(p+k)
-c(2p+k)][ (n+ l)BIP+ 1 + (n-l>(2~.~+k)A~~-~+(n+1)(2~+$_)A,r-”-~ +&($p:y) -M(n+
[&I(~+
{
[ML(W
- (n-l)K&WI
+hJ,(W
1
4n(~+k)[(n+l)B,r”-‘+(n-l)B,r-“-‘I -i
m
l)B,r-“1
(~-l)~n-I(Wll}
l)Zn+@)--
t&.= t,,+ik$y mrz=iy
l)&+I(i+)
(n-
k(ZG+l(~r)
+ZL(&))
-A,(Z,+,(&)
+I.l(sr))]}
4n(~+k)[(n+l)BI~-1-(n-l)B,r-n-1]+~[A,K,(Gr)+A,Z,(6r)]
(67)
where
t?T=
2cA+2(p+k)
+c(2p+k) 1+c
-2(2~+k)A,r-3S~~~~=~~
A r_,+-8cA+4(p+k) 2
{A,&(&)
-6c(2p+k) I-3c
--A,Z,(Sr)}
A r 3
1006
B. M. CHIU
t*e=2ch-2(/.h+k)+c(2p+k)A l+C i2(2p-t-k)A4r-3* tro = A i
c(2 I
and JAMES
D. LEE
r_,+-8cX-4(p+k)-2c(2p+k)A 2 l-3c ~~~~~~~ wG(Sr)
-tk) 4(p+k) l’+ c &r-l -t
r 3
-N,(Sr)I
-2c(2p+k) l-3c
/g r+2(2p+k)A 3
4
r-3 I
m et=-Y
(68)
t2=0 trr=
(2hf2,~+k)A,-(2p+k)A~r~~
tee= (2h-b2p+k)AI+
(2p+k)A,re2
tM = i - (2p.+k)A4r++ II (Sr) -r-f-
ter = i - (2~ + k)A4rp2 +
172, =
~y{S[--~5~dSr)
lo(Sr)
+4J@r)l}
m @r--0
(69) 4. STRESS
FUNCTIONS
Mindlin[5,6] introduced two stress functions for his couple-stress theory in the same fashion as the Airy stress function was introduced in classical elasticity. His fo~ulation can be modified slightly to fit the micropolar theory (cf. Kaloni and Ariman [4]). We now reproduce Mind&r’s results in our notations. The relations between stress functions and components of stress and moment stress are (70) (71) (72)
On the plane problem in micropolar elasticity
1007
ter =I -iF,t-e+$F,e+Gw
(731
mrr= GW
(74)
I G,e.
mb=;
(75)
And the governing equations for the stress functions are (761 (77)
V4F=#,V2
G-~,G)=O. C
(78)
The general solutions for F and G, co~esponding to the soluti~us found in the previous section, are
Case2:
n=+l
F = ekze (2pfk)A,r’+
CaseJ:
&A*[
(*p-i-k)
(-rlnr)
-2i(h+2p+k)rOl
n=O G = iy[A,K,(6r)
F=$
(2h+2p+k)AIrZ
5. EXCEPTIONAL
+A,I,(6r)]
(83)
-(2pfk)A21nr-i(2p+k)A&
034)
CASES-VOLTERRA’S
DISLOCATIONS
In the previous sections, we have formulated the static problem of micropolar elasticity in cylindrical coordinates and obtained general solutions for boundary value
1008
B. M. CHIU
and JAMES
D. LEE
problems of the first and second kinds in a circular ring-shaped region. However those solutions are only valid for single-valued displacement and microrotation fields. In this section, we demonstrate the physical meaning of many-valued solutions by giving the following two examples: Example
1
We cut a ring (see Fig. 2) and apply forces P until an initial displacement d is obtained. Then we rejoin the two ends by welding or other means. A ring with initial stresses is thus obtained. P
I
Fig. 2. Dislocation of ring with parallel fissure.
We try the following set of solutions for this problem, r/G-,@ =G v(r,fQ =G c$(r,O)
d
sin 0 -ecoso+4(h+2P+k)
d
cos 8 ~sin~+4(h+2P+k)
[2(2~++) [2(2p+k)
lnrlnr+
(2X+2~+$_)] (2h+21.~+k)]
(85) (86)
=$yc
Substituting equations (85-87) into equations (9-l l), we realize that the equations of motion are satisfied identically. From equations (12- 17), we obtain t,,=
Iss=*sine 2~
h=f0r=-~
mrz =m0.:=-y--
G%+$-~)(~A+~P+N 2(h+2/_+k)
r
d cosO (2/.~+k) (2h+2p+k) r
2(h+2p+k)
d cos 0 2rr r2 .
In order to have stress free boundaries, i.e. t,, = trO= m, = 0 at r = R,, and r = Ri, another set of solutions (making use of the results obtained in section 3)
we superpose
1009
On the plane problem in inicropolar elasticity
and we get u(r,8)
=gsinO
2h-2 -k 2p fk
B8 + BP
1 1 2(2p+kk) +2p+k
II Br2_B,__Z 1 tr(r,O) =&se i6A+5(2P+k) x B Kt(Sr) -&(‘r) I 5 2
_B
6
Zz(6r) -Zo(sr) 2
a
4
--
2p.+k
x B 5 &(‘r) i
+&far) 2
2(2p+k)
_B 6 zdSr) +f,t6r) 2
(89)
1 2p+k
I}
(90)
where B3 = M/N B = (2p+k) (2h+2p+k) 4 2(h+2,u+k) -
Rf_2(2h+2p+k)
& (S&l Wm -R&J I+
~6~(~+2~+k)~ (2p+k)SN
x [ I2 (SR,) Rf - Zz(SR,) R:] } + N = 2(2h-t2p+k)S(R~--RI)
~Rf+~[Zp(6Ri)(R19*-R:h) R% & (SRd (rlz - 52)
(2~+k)~2~+2~+k) 2(h+2p+k)
- ‘6y(:;f:~+k)
ScRz_RR2)
’ { ([1-ql)
O
[Zz(&Ri)Rf
51 G Kz(SRi) +Ko(6Ri) 362 5 Zz(SRi) +Zo(6Ri) rll = &(SRo)
+ Ko(6Ro) t 72 = Zz(6R,,) + Z,(6Ro).
In this case a portion of the ring between two adjacent cross sections is cut out (see Fig. 3). The ends of the ring are joined again by welding or other means. We again obtain a ring with initial stresses.
1010
B. M. CHIU
and JAMES
D. LEE
Fig. 3. Dislocation of ring with radial fissure.
For this problem we try the following set of solutions: 2p+k
“Cry e,=4n(X+2p+k)
vlnr
(92)
U(T, e) =+I $(r, e)
(93)
=&e.
Again we find that this set of solutions satisfies equations (9- 11). Correspondingly, obtain
2h+2p-+k (2p+k)2(h+2p+k)
h(2E.c+k)
lnr+2(A+2p+k)
we
+A+2p+k
fro = ter = mrz = 0
m BZ
=zf;;y+.
(95)
Making use of the results obtained in section 3, we superpose another set of solutions: U(T, 0) = AIr+Az~-l,
u(r, 0) = $(r, e) = 0.
(96)
By taking
(97)
On the plane problem in micropolar elasticity
1011
we have made t,, = fro= 171,= 0 at r = R,, and r = Ri. It is interesting to note that this ring has the following initial stresses: E (2/L+k)(2h+2p+k) tee = G h+2/_L+k
(2+ln(&)+~ln(~)[R~(l-$!
+R;(l-$)I} (98) These stresses are B-independent. 6. CONCLUSION
The main purpose of the present work is to solve the boundary value problem of micropolar elasticity in a complete ring-shaped region. At the boundaries r = Ri and r = Ro, either displacements and microrotation U, o, + or stress and moment stress components trr, rM, m,, are specified as functions of 8. For single-valued solutions, U, u, and 4 are periodic in 8 with period of 2~. Therefore the set {efne;n = 0, n = + 1, n = 52,. . . .} forms a complete orthonormal basis. We then reduce the problem to a system of three simultaneous ordinary differential equations of second order with the Fourier coefficients U,(r), V,(r), and an(r) being the unknowns. Finally we obtain the most general solutions of U,, V,, and @‘nfor every n with six arbitrary constants. The remaining task for each boundary value problem is to seek the Fourier coefficients of given boundary conditions and then determine the corresponding six constants according to the given boundary conditions. We also give two examples to show the physical meaning of many-valued solutions and illustrate how the corresponding single-valued solutions are being utilized. Acknowledgement-The second author (JDL) is indebted to the support by a grant, No. GK-20657, from National Science Foundation. REFERENCES [l] A. C. ERINGEN and E. S. SUHUBI, Inr. J. Engng Sci. 2,189 (1964). [2] A. C. ERINGEN,J. Math. Mech. 15,909 (1966). [3] A. C. ERINGEN,J. Math. Mech. 16,1(1966). [4] P. K. KALONI and T. ARIMAN,ZAMP 17,136 (1967). [5] R. D. MINDLIN and H. F. TIERSTEN,Arch. ration Mech. Anafysisll, 415, (1962). [6] R. D. MINDLIN, Exp. Mech. 3,1(1963). [7] T. ARIMAN,Acta Mech. 4,216 (1967). [8] H. K. PARHI and A. K. DAS, Bull. Acnd. Pofon. Sci.,Ser. Sci. Tech. 19,13 1 (1971). 191 R. V. CHURCHILL, Fourier Series and Boundary Value Problems. McGraw-Hill (1963). 1101F. B.HILDEBRAND,Aduanced CafculusforAppfications. Prentice-Hall (1962). (Received 14 September 1972) Resume- La solution la plus genkale de I’ClasticitC micropolarie a &tCobtenue pour des problemes statiques plans darts une region circulaire en forme d’anneau. A titre d’illustration, deux exemples de probltme de dislocation de Volterra sont don&s pour montrer la signification physique des solutions a plusieurs valeurs. Zusammenfassung - Die allgemeinste L&sung mikropolarer Elastixitit wurde fiir statische ebene Probleme in einer runden ringfkmigen Zone erhalten. Zur Verbildlichung werden xwei Beispiele von Volterra’s Verdr@tmgsproblem gegeben, urn die physikalische Bedeutung mehrfach-bewerteter Losungen nachzuweisen.
1012
B. M. CHIU
and JAMES
D. LEE
Sommario - La soluzione pili generale dell’elasticit8 micropolare t stata ricavata nei riguardi di problemi di piano static0 in una regione circolare a forma di cerchio. A scopi illustrativi, si danno due esempi de1 problema di spostamento di Volterra per dimostrare il significato fisico delle soluzioni a pi6 valori. A~IZT~~KT - nonyreHo Hau6onee 06mee pemeHue ME~K~OIIOJIX~HO~~ YIIPY~OCTH AJIHCTaTn’IecKnx ~OCKAX npo6neM B KOJIbueO6pa3HOiio6nacTn. B qeJuix HarJt~AHOCTA,naIOTCnnBe npuMepa npO6neMbI ~ncnoKaqm% BOJlbTeppbl nOKa3aTb @i3WieCKHi CMblCJIMHOr03HaYHbIX npo6neiw